Degenerate Regge trajectories and parity doublets of baryon resonances

Volume 26B, n u m b e r 2

PHYSICS

IETTERS

25 D e c e m b e r 1967

DEGENERATE REGGE TRAJECTORIES AND DOUBLETS OF BARYON RESONANCES*

PARITY

V. BARGER and D. C L I N E Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA Received 19 September 1967

The MacDowell reflection s y m m e t r y as applied to Regge r e c u r r e n c e s of SU(3) multiplets is suggested as a natural explanation of r e c e n t e x p e r i m e n t a l observations of parity doublets of baryon r e s o n a n c e s (resonance p a i r s with approximately equal m a s s e s , the s a m e Y, I and J values and opposite parities).

A c c u m u l a t i n g e x p e r i m e n t a l e v i d e n c e [1] s u p p o r t s t h e e m p i r i c a l h y p o t h e s i s [2] t h a t t h e m a j o r i t y of t h e k n o w n Y = + 1 b a r y o n s t a t e s l i e o n Regge trajectories that rise linearly with (mass~. E x t e n s i o n of t h i s b a r y o n r e c u r r e n c e c l a s s i f i c a t i o n s c h e m e to i n c l u d e 1, 8 a n d 10 SU(3) m u l t i p l e t s o n linear trajectories also seems quite promising [2, 3]. In t h i s n o t e we p r e s e n t f u r t h e r e v i d e n c e for linear baryon trajectories based on recent e x p e r i m e n t a l d i s c o v e r i e s [ 3 - 6 ] of s e v e r a l r e s onance parity doublets (resonance pairs with approximately degenerate masses, the same Y,I and J values, and opposite parities). Sensitive e x p e r i m e n t a l t e s t s of t h e a p p r o x i m a t e l i n e a r n a t u r e of t h e t r a j e c t o r i e s a r e d i s c u s s e d a l o n g with the implications for further classifications of b a r y o n r e s o n a n c e s . T h e f o u n d a t i o n of o u r c o n s i d e r a t i o n s i s t h e M a c D o w e l l r e f l e c t i o n s y m m e t r y [7] a s a p p l i e d to R e g g e r e c u r r e n c e s [2, 8, 9] of SU(3) m u l t i p l e t s . A c c o r d i n g l y , in p s e u d o s c a l a r m e s o n - n u c l e o n s c a t t e r i n g t h e e x i s t e n c e of a R e g g e p o l e i n t h e j t h p a r t i a l w a v e a m p l i t u d e of s i g n a t u r e 7 a n d p a r i t y P r e q u i r e s a c o r r e s p o n d i n g R e g g e p o l e in t h e j t h p a r t i a l w a v e a m p l i t u d e of s i g n a t u r e 7 a n d parity-P. T h u s , if [8] J

h p ~ W ) = j _ a~(w) (W)

'

(1)

then

hJ %- F /

=J - ~(W)

(2)

* Work supported in p a r t by the University of Wisconsin R e s e a r c h Committee with funds granted by the Wisconsin Alumni R e s e a r c h Foundation, and in p a r t by the U. S. Atomic Energy Commission u n d e r contract AT(11-1)-881 #COO-881-116.

where the trajectories and residues are related by reflection in the Wplane:

Y(W) =a(W)

,

~(W) = - ~ ( - W ) .

(3) (4)

In t h e c i r c u m s t a n c e t h a t R e a(W) ~- R e a ( - W ) f o r R e W > 0, I m W = 0, r e c u r r e n c e s a l o n g t h e a(W) and a(W) trajector~s willappear as resonance d o u b l e t s w i t h M ~-M, J = J a n d P = - f t . S u c h a c o r r e l a t i o n of p a i r s of p o s i t i v e a n d n e g a t i v e p a r i t y resonances with approximately equal mass and t h e s a m e J s e e m s to b e i m p l i e d b y t h e p r e s e n t experimental situation as will be discussed below. F o r t h e r e g i o n W = 1 to 3 GeV t h e k n o w n b a r y o n r e s o n a n c e s p e c t r a s e e m to b e w e l l r e p r e s e n t e d b y t r a j e c t o r i e s of t h e s i m p l e f u n c t i o n a l f o r m [2] R e a(W) = a + c W 2

(5)

R e c e n t d i s c o v e r i e s of r e s o n a n c e s i n t h e m a s s r e g i o n W = 3 to 4 GeV c o n t i n u e to i n d i c a t e c o n s i s t e n c y w i t h t h i s a p p r o x i m a t e f o r m ~. H o w e v e r , it i s s t i l l p o s s i b l e to c o n s t r u c t f u n c t i o n s t h a t r o u g h l y r e p r o d u c e eq. (5) f o r R e W > 0, I m W = 0 while deviating significantly from this form for R e W < 0, I m W = 0. T h e v a l i d i t y of t h e r e p r e s e n t a t i o n i n eq. (5) in t h e r e g i o n R e W < 0 c a n b e e v a l u a t e d t h r o u g h t h e s y m m e t r y r e l a t i o n of eq. (3) b y a s s i g n m e n t of r e s o n a n c e s to t h e a'(W) t r a j e c tory. T h e p r e s e n t s t a t u s of p a r i t y d o u b l e t s on M a c D o w e l l s y m m e t r i c t r a j e c t o r i e s f o r t h e (a,fl) o c t e t

~- Ballam et al. [1] have r e p o r t e d p r e l i m i n a r y evidence for I = ½ r e s o n a n c e s at 3340, 3660 and 3960 MeV which lie on a l i n e a r extrapolation of the Ny t r a j e c t o r y . 85

Volume 26B. n u m b e r 2

PHYSICS

i s d i s p l a y e d in fig. 1 * * . T h e f o l l o w i n g r e c e n t experimental discoveries are pertinent for this (a,/3) o c t e t r e c u r r e n c e c l a s s i f i c a t i o n : i) NB(1670;-~-) f r o m nN p h a s e s h i f t a n a l y s e s [3], ii) A ~ ( 1 8 2 7 ; ~ ) f r o m a K - p ~ ~ p h a s e s h i f t a n a l y s i s [4], iii) ~ ( 2 2 6 0 ) a n d A(2340) f r o m ~ total crosss e c t i o n s [3]. F o r t h e s t a t e s of k n o w n s p i n - p a r i t y , t h e a s s i g n m e n t s in fig. 1 a r e u n a m b i g u o u s a n d p r o v i d e reasonably strong support for linear trajectories of t h e f o r m g i v e n in eq. (5). A t e n t a t i v e a s s i g n m e n t of ~ ( 2 2 6 0 ) to t h e ~/3 t r a j e c t o r y a n d t h e A(2340) to t h e A a t r a j e c t o r y i s m a d e o n t h e b a s i s of e l a s t i c i t y c o n s i d e r a t i o n s (i. e. b o t h ~fi(1765, y)5a n d ~ ( 2 2 6 0 ) h a v e r e l a t i v e l y l a r g e e l a s t i c i t i e s [3] c o m p a r e d to o t h e r ~, r e s o n a n c e s in s i m i l a r m a s s r e g i o n s ; s i m i l a r l y t h e A a ( 1 8 2 7 ; ~ - ) a n d A(2340) r e s o n a n c e s a r e r e l a t i v e l y e l a s t i c [3]. Tentative conclusions that can be drawn from t h e s e (a,/3) o c t e t c l a s s i f i c a t i o n s a r e : (a) t h e a p p r o x i m a t e e q u a l i t y R e a(W) ~- R e ~ ( W ) s p e a k s a g a i n s t t h e a p p e a r a n c e of a s t r o n g l i n e a r t e r m in W. T h u s we a r e l e d to e x p e c t t h a t t h e h i g h e r m a s s s t a t e s of t h e (a,/3) o c t e t s w i l l a l s o f o l l o w a t r a j e c t o r y of t h e _ f o r m of eq. (5); ( b ) t h e a p p a r e n t n o n e x i s t e n c e of ½ s t a t e s w i t h m a s s M ~- 1 GeV p u t s strong constraints on the residue function: /3(M) = 0, o r e q u i v a l e n t l y , /3(-M) = 0 ~ ; (c) t h e m a s s i n v e r s i o n of t h e ~/3(1760, ~-) a n d A f i ( 1 8 3 0 , ~ - ) s u g g e s t s a s i m i l a r i n v e r s i o n $~ f o r ~ a ( ? , -~ +) a n d ** S i m i l a r suggestive evidence exists for parity doubling of the (T,6) octet multiplets (-~-, ~+). The limited space available here does not allow a detailed discussion of this possibility. However see refs [5,6]. j" A sign change of the Not residue/3 (W) between W + M N and W : ~ M i s in fact required to avoid a ghost state at M(cf. Desai [8]7. However, in this classification s c h e m e the p r e c i s e r e a s o n for the kinematical or dynamical z e r o at M, which extinguishes the ½- bound state, r e m a i n s unknown. An a l t e r n a t i v e possibility is to r e g a r d the [N(1570, ½-), A(1670, ~--), ~(1780, ½-), ~(?)] as the m i s s i n g ½- s t a t e s on the fi t r a j e c t o r i e s . These a s s i g n m e n t s would r e s u l t in very nonlinear f o r m s for the /3 t r a j e c t o r i e s as could be seen f r o m fig. 1. In the p r e s e n t t r e a t m e n t we have p r e f e r r e d to r e g a r d these ½~/-baryon states as threshold e n h a n c e m e n t s with no higher r e c u r r e n c e pattern. However if the ?/-baryon s t a t e s do lie on new Regge t r a j e c t o r i e s , the Roper resonance N(1450. 21+) may be the MacDowell reflection of the N(1570, ½-) state. -~-~In the past the ~(1910) has been assigned as ~ot(1910, ~+). At p r e s e n t no convincing s p i n - p a r i t y d e t e r m i n a t i o n s exist for this resonance. In addition the b r a n c h i n g ratios for the 1910 resonance are inconsistent with the ot-octet a s s i g n m e n t [10]. We 86

LETTERS

25 D e c e m b e r 1967

A n ( 1 8 1 0 , ~+). T h e i n v e r s i o n l e a d s to a v a l u e of a2(0) much less than aA(0) which should have observable consequences in the energy dependence of r e a c t i o n s d o m i n a t e d b y ~'a a n d A a e x c h a n g e s ~$$. The particle mass spectra can only determine R e a(W) a n d fl(W) f o r W2 > 0. H o w e v e r , t h e r e g i o n W2 < 0 c a n b e s t u d i e d f r o m b a c k w a r d p s e u d o - s c a l a r m e s o n - n u c l e o n s c a t t e r i n g . In principle the trajectory and residue function can be analytically continued from the region W 2 > 0 to W 2 < 0. In p r a c t i c e a w o r k i n g h y p o t h e s i s i s to a s s u m e t h a t t h e s i m p l e f u n c t i o n a l f o r m s f o r R e a(W) a n d fi(W) d e t e r m i n e d f r o m t h e p a r t i c l e s p e c t r a a r e a l s o v a l i d r e p r e s e n t a t i o n s f o r W2 < 0. T h e p r e c e d i n g d i s c u s s i o n of t h e M a c D o w e l l s y m m e t r i c t r a j e c t o r i e s a f f o r d s a d i r e c t t e s t of t h i s h y p o t h e s i s . N a m e l y , f o r W 2 < 0, He a(W) s h o u l d h a v e t h e f o r m of eq. (5) a n d fi(W) s h o u l d r e f l e c t t h e a b s e n c e of t h e l o w e s t l y i n g s t a t e s o n t h e / 3 o c t e t (½- s t a t e s ) . We now s p e c i f i c a l l y d i s c u s s t h e r e s i d u e f u n c t i o n f o r t h e N a t r a j e c t o r y [9]. A p o s s i b l e r e presentation for/3(W) which satisfies the requirem e n t s R e a ( W ) -~ R e a(W) a n d ~ ( M ) = 0 i s

/3(w) -- (1 + M ~ ) ~n(W) ,

(8)

w h e r e M N i s t h e n u c l e o n m a s s a n d N/ -~ M N. In o r d e r to t e s t t h e v a l i d i t y of t h e r e p r e s e n t a t i o n of eq. (8) f o r W 2 < 0, we h a v e a n a l y z e d a l l a v a i l a b l e d a t a o n b a c k w a r d nJ:p e l a s t i c s c a t t e r i n g u s i n g l i n e a r t r a j e c t o r i e s in W2 a n d r e s i d u e f u n c t i o n s f o r t h e N a a n d •5 R e g g e p o l e s of t h e f o r m

~(w) = (1 + ~w)~

(9)

where the parameters (5,7) for each trajectory w e r e t r e a t e d a s v a r i a b l e c o n s t a n t s i n ×2 f i t t i n g p r o g r a m . A t m i n i m u m ×2, t h e s o l u t i o n o b t a i n e d f o r t h e N a r e s i d u e w a s [11] 5 N ~ 1.07//I//N ,

(10)

c o r r e s p o n d i n g t o a z e r o of/3 at W ~ - 0 . 9 3 M N , i n a c c o r d w i t h t h e r e s u l t s of eq. (8) e x p e c t e d f r o m 1the e x p e r i m e n t a l a b s e n c e of ~ s t a t e s i n t h e W2 > 0 r e g i o n . T h i s r e s u l t i s a p r e l i m i n a r y i n speculate that the 1910 may be the ~ ( 1 9 1 0 . ~-) decuplet m e m b e r which is the MacDowell reflected Z5(2035, ~+) resonance. The branching ratios of the ~ (1910) are suggestive of a decuplet a s s i g n m e n t as are the branching r a t i o s of the ~6(2035, ~+). -~"~ Since backward production in the reaction n - p Y O (15207 K must proceed by octet ~ .exchange [assuming a singlet a s s i g n m e n t for Y~(1520)], we expect a rapid d e c r e a s e with energy for this differential c r o s s - s e c t i o n n e a r 180 ° . Similar conclusions follow for 7r-p ~Yo(1405) K +.

Volume 26B, number 2

PHYSICS

B

LETTERS

OCTET TRAJECTORIES|~-I

2

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25 D e c e m b e r 1967

OCTET T R A J E C T O R I E S

2

f

t

rlr,

rr

-1.

/

;'1":

z

KINEMATICAL

2

OR

DYNAMICAL ZEROS OF R E G G E R E S I D U E S

6

5

4

3

2

\r~\\l

W2

//,V .

2

3

4

5

--~

W z-

Fig. 1. P o s s i b l e a s s i g n m e n t s of baryon r e s o n a n c e s to parity doublets on the MacDowell s y m m e t r i c (~,/3) octet t r a j e c t o r i e s . d i c a t i o n of t h e t y p e of t e s t s t h a t c a n b e m a d e f r o m a n a l y s e s of d a t a o n b a c k w a r d m e s o n - b a r y o n scattering. In s u m m a r y , w e h a v e s k e t c h e d a t e n t a t i v e classification scheme for the observed parity doublets which is also c o n s i s t e n t with p r e s e n t d a t a on b a c k w a r d p s e u d o s c a l a r m e s o n - n u c l e o n s c a t t e r i n g . T h e g r o w i n g n u m b e r of n e g a t i v e p a r i t y s t a t e s fit n a t u r a l l y i n t o t h i s c l a s s i f i c a t i o n with t r a j e c t o r i e s l i n e a r in W 2, a n d h e n c e M a c D o w e l l s y m m e t r i c . T h e s p e c i f i c c o n s e q u e n c e of t h i s s c h e m e i s t h e p r e d i c t i o n o f a n u m b e r of n e g a t i v e ( p o s i t i v e ) p a r i t y r e s o n a n c e s , e a c h in t h e m a s s v i c i n i t y of a k n o w n p o s i t i v e ( n e g a t i v e ) p a r i t y r e s o n a n c e . A g r e a t d e a l of e x p e r i m e n t a l e f f o r t will u n d o u b t e d l y b e r e q u i r e d to d i s e n t a n g l e t h e s e doublets. 1. J. Ballam et a l . , Bull. Am. Phys. Soc. 12 (1967) 13Fl1: J. Bartke et al. , Phys. L e t t e r s 2413 (1967) 118. 2. V. B a r g e r and D. Cline, Phys. Rev. 155 (1967) 1792:

3. 4. 5.

6.

7. 8. 9. 10. 11.

Phys. Rev. L e t t e r s 16 (1966) 913. The notation is defined in these r e f e r e n c e s . A.H. Rosenfeld et a l . , Rev. Mod. Phys. 39 (1967) 1. R. A r m e n t e r o s et a l . , Phys. L e t t e r s 24B (1967) 198, r e p o r t discovery of A/3(1827 :~-). A.D. Brody et a l . , Bull. Am. Phys. Soc. 12 (1967) 3+ BF15 have suggested a possible N5(1620, ~ ) resonance which decays predominantly into A5(1238 ) + 7/. M. D e r r i c k et a l . , Phys. Rev. L e t t e r s 18 (1967) 266; D. C. Colley et a l . , Phys. L e t t e r s 2413 (1967) 489; J. Mott, Northwestern University thesis (1967). These authors p r e s e n t evidence for F~5(1680 , 3+?). S, W. MacDowell, Phys. Rev. 116 (1959) 774. V. Singh, Phys. Rev. 129 (1963) 1889: I. A. Sakmar, Phys. Rev. 135 (1964} B249: B. R. Desai, Phys. Rev. L e t t e r s 17 (1966) 498. C.B. Chiu and J, D. Stack, Phys. Rev. 153 (1967) 1575. R. Tripp et a l . , UCRL-17385 Rev. (1967). V. B a r g e r and D. Cline, to be published. Recent data on backward 7r±p elastic scattering in the momentum range 6 to 17.5 GeV/c w e r e included in this analysis. [See C. J. S. Damerell et a l . , Phys. Rev. L e t t e r s 19 (1967) 460]. 87